g08rbf calculates the parameter estimates, score statistics and their variance-covariance matrices for the linear model using a likelihood based on the ranks of the observations when some of the observations may be right-censored.
The routine may be called by the names g08rbf or nagf_nonpar_rank_regsn_censored.
Analysis of data can be made by replacing observations by their ranks. The analysis produces inference for the regression model where the location parameters of the observations, , for , are related by . Here is an matrix of explanatory variables and is a vector of unknown regression parameters. The observations are replaced by their ranks and an approximation, based on Taylor's series expansion, made to the rank marginal likelihood. For details of the approximation see Pettitt (1982).
An observation is said to be right-censored if we can only observe with . We rank censored and uncensored observations as follows. Suppose we can observe , for , directly but , for and , are censored on the right. We define the rank of , for , in the usual way; equals if and only if is the th smallest amongst the . The right-censored , for , has rank if and only if lies in the interval , with , and the ordered , for .
The distribution of the is assumed to be of the following form. Let , the logistic distribution function, and consider the distribution function
defined by . This distribution function can be thought of as either the distribution function of the minimum,
, of a random sample of size from the logistic distribution, or as the being the distribution function of a random variable having the -distribution with and degrees of freedom. This family of generalized logistic distribution functions naturally links the symmetric logistic distribution with the skew extreme value distribution () and with the limiting negative exponential distribution (). For this family explicit results are available for right-censored data. See Pettitt (1983) for details.
Let denote the logarithm of the rank marginal likelihood of the observations and define the vector by , and let the diagonal matrix and symmetric matrix be given by . Then various statistics can be found from the analysis.
(a)The score statistic . This statistic is used to test the hypothesis (see (e)).
(b)The estimated variance-covariance matrix of the score statistic in (a).
(c)The estimate .
(d)The estimated variance-covariance matrix of the estimate .
(e)The statistic , used to test . Under , has an approximate -distribution with degrees of freedom.
(f)The standard errors of the estimates given in (c).
(g)Approximate -statistics, i.e., for testing . For , has an approximate distribution.
In many situations, more than one sample of observations will be available. In this case we assume the model,
where ns is the number of samples. In an obvious manner, and are the vector of observations and the design matrix for the th sample respectively. Note that the arbitrary transformation can be assumed different for each sample since observations are ranked within the sample.
The earlier analysis can be extended to give a combined estimate of as , where
with , and defined as , and above but for the th sample.
The remaining statistics are calculated as for the one sample case.
Kalbfleisch J D and Prentice R L (1980) The Statistical Analysis of Failure Time Data Wiley
Pettitt A N (1982) Inference for the linear model using a likelihood based on ranks J. Roy. Statist. Soc. Ser. B44 234–243
Pettitt A N (1983) Approximate methods using ranks for regression with censored data Biometrika70 121–132
1: – IntegerInput
On entry: the number of samples.
2: – Integer arrayInput
On entry: the number of observations in the
th sample, for .
, for .
3: – IntegerInput
On entry: the total number of observations.
4: – Real (Kind=nag_wp) arrayInput
On entry: the observations in each sample. Specifically, must contain the th observation in the th sample.
5: – IntegerInput
On entry: the number of parameters to be fitted.
6: – Real (Kind=nag_wp) arrayInput
On entry: the design matrices for each sample. Specifically, must contain the value of the th explanatory variable for the th observations in the th sample.
must not contain a column with all elements equal.
7: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g08rbf is called.
8: – Integer arrayInput
On entry: defines the censoring variable for the observations in y.
If is uncensored.
If is censored.
or , for .
9: – Real (Kind=nag_wp)Input
On entry: the value of the parameter defining the generalized logistic distribution. For , the limiting extreme value distribution is assumed.
10: – IntegerInput
On entry: the value of the largest sample size.
11: – Real (Kind=nag_wp)Input
On entry: the tolerance for judging whether two observations are tied. Thus, observations and are adjudged to be tied if .
12: – Real (Kind=nag_wp) arrayOutput
On exit: the variance-covariance matrices of the score statistics and the parameter estimates, the former being stored in the upper triangle and the latter in the lower triangle. Thus for , contains an estimate of the covariance between the th and th score statistics. For , contains an estimate of the covariance between the th and th parameter estimates.
13: – IntegerInput
On entry: the first dimension of the array prvr as declared in the (sub)program from which g08rbf is called.
14: – Integer arrayOutput
On exit: for the one sample case, irank contains the ranks of the observations.
15: – Real (Kind=nag_wp) arrayOutput
On exit: for the one sample case, zin contains the expected values of the function of the order statistics.
16: – Real (Kind=nag_wp) arrayOutput
On exit: for the one sample case, eta contains the expected values of the function of the order statistics.
17: – Real (Kind=nag_wp) arrayOutput
On exit: for the one sample case, vapvec contains the upper triangle of the variance-covariance matrix of the function of the order statistics stored column-wise.
18: – Real (Kind=nag_wp) arrayOutput
On exit: the statistics calculated by the routine.
The first ip components of parest contain the score statistics.
The next ip elements contain the parameter estimates.
contains the value of the statistic.
The next ip elements of parest contain the standard errors of the parameter estimates.
Finally, the remaining ip elements of parest contain the -statistics.
19: – Real (Kind=nag_wp) arrayOutput
20: – IntegerInput
21: – Integer arrayOutput
On entry: are no longer required by g08rbf but is retained for backwards compatibility.
22: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, elements of .
On entry, .
On entry, .
On entry, and .
On entry, and .
On entry, and .
On entry, and .
On entry, .
On entry, .
On entry, and .
On entry, elements of icen are out of range.
Constraint: or , for all .
On entry, all the observations were adjudged to be tied. You are advised to check the value supplied for tol.
The matrix is either ill-conditioned or not positive definite. This error should only occur with extreme rankings of the data.
On entry, for , for all .
Constraint: for at least one .
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
The computations are believed to be stable.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g08rbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g08rbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The time taken by g08rbf depends on the number of samples, the total number of observations and the number of parameters fitted.
In extreme cases the parameter estimates for certain models can be infinite, although this is unlikely to occur in practice. See Pettitt (1982) for further details.
This example fits a regression model to a single sample of observations using just one explanatory variable.